The sale load, $L$ of a wooden bearo of width $w$, height $h$, and length $L$ supported at both ends, varies directly as the product of the width and the square of the height and inversely as the length. A wooden beam 4 inches wide, 8 inches high, and 216 inches long can hold a load of 6390 pounds. What load would a beam 2 inches wide, 6 inches high, and 144 inches long, of the same material support? Round your answer to the nearest integer if necessary

Answer
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pounds

Step 1 ：Given that the load, $L$ of a wooden beam of width $w$, height $h$, and length $L$ supported at both ends, varies directly as the product of the width and the square of the height and inversely as the length. This can be represented by the equation $L = k \cdot \frac{w \cdot h^2}{L}$, where $k$ is a constant of proportionality.

Step 2 ：We are given that a wooden beam 4 inches wide, 8 inches high, and 216 inches long can hold a load of 6390 pounds. We can use these values to find the value of $k$. Substituting these values into the equation, we get $6390 = k \cdot \frac{4 \cdot 8^2}{216}$. Solving for $k$, we get $k = 5391.5625$.

Step 3 ：We are asked to find the load that a beam 2 inches wide, 6 inches high, and 144 inches long, of the same material can support. Substituting these values and the value of $k$ into the equation, we get $L = 5391.5625 \cdot \frac{2 \cdot 6^2}{144}$. Solving for $L$, we get $L = 2696$.

Step 4 ：Final Answer: The load that a beam 2 inches wide, 6 inches high, and 144 inches long, of the same material can support is \(\boxed{2696}\) pounds.